Considering a Classical Upper Bound on the Frobenius Number
Abstract
:1. Introduction
2. Preliminary and Auxiliary Results
3. Observations on a Previously-Known Upper Bound
4. Tightness Comparison of Upper Bounds
5. Proof of Theorem 1
6. Proof of Theorem 2
- if , then ;
- if , then ;
- if , then ;
- if , then ;
- if , then .
- : In this case, notice that since the entries of the vector are coprime by assumption, it follows that we have and . Note that we have and . These inequalities imply thatFinally, observe that the equality holds for all and, thus, is a valid upper bound in this scenario.
- : In this case, notice that
- : In this case, notice that
- : In this case, notice that
- : In this case, notice that
7. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Williams, A.; Haijima, D. Considering a Classical Upper Bound on the Frobenius Number. Mathematics 2024, 12, 4029. https://doi.org/10.3390/math12244029
Williams A, Haijima D. Considering a Classical Upper Bound on the Frobenius Number. Mathematics. 2024; 12(24):4029. https://doi.org/10.3390/math12244029
Chicago/Turabian StyleWilliams, Aled, and Daiki Haijima. 2024. "Considering a Classical Upper Bound on the Frobenius Number" Mathematics 12, no. 24: 4029. https://doi.org/10.3390/math12244029
APA StyleWilliams, A., & Haijima, D. (2024). Considering a Classical Upper Bound on the Frobenius Number. Mathematics, 12(24), 4029. https://doi.org/10.3390/math12244029